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Zorluk: OrtaRatios, Rates, and Proportions

A community orchard initially planted apple, peach, and plum trees in the ratio 4:3:54 : 3 : 5, respectively. After a severe frost, 1212 plum trees died and were removed from the orchard, while all apple and peach trees survived. If the new ratio of apple trees to remaining plum trees became 2:12 : 1, how many total trees were originally in the orchard?

  1. A
    2020
  2. B
    3636
  3. C
    4040
  4. 4848Cevap
  5. E
    6060

Cevap

The total number of trees originally in the orchard was 48.
By representing the initial tree counts with the multiplier xx as 4x4x (apple), 3x3x (peach), and 5x5x (plum), the total original tree count is 12x12x. Setting up the proportion 4x5x−12=21\frac{4x}{5x - 12} = \frac{2}{1} gives 4x=10x−244x = 10x - 24, which simplifies to 6x=246x = 24, or x=4x = 4. Multiplying x=4x = 4 by the total ratio units (1212) yields 4848 original trees.

Adım Adım Çözüm

1
Define variables using the initial ratio.
Let the original number of apple, peach, and plum trees be 4x4x, 3x3x, and 5x5x, respectively. The original total number of trees is 4x+3x+5x=12x4x + 3x + 5x = 12x.
Ratios allow quantities to be represented in terms of a common multiplier xx.
2
Set up an equation based on the updated number of plum trees and the new ratio.
After removing 1212 plum trees, the number of plum trees is 5x−125x - 12. The ratio of apple trees to remaining plum trees is 4x5x−12=21\frac{4x}{5x - 12} = \frac{2}{1}.
The number of apple trees remains 4x4x, and the problem states the new apple-to-plum ratio is 2:12 : 1.
3
Solve the proportion for xx.
4x=2(5x−12)  ⟹  4x=10x−24  ⟹  6x=24  ⟹  x=44x = 2(5x - 12) \implies 4x = 10x - 24 \implies 6x = 24 \implies x = 4.
Cross-multiplying eliminates fractions and isolates the variable xx.
4
Calculate the original total number of trees.
Original total trees =12x=12(4)=48= 12x = 12(4) = 48.
Substituting x=4x = 4 back into the original total expression yields the requested answer.

Anahtar Kavram

Solving multi-part ratio problems with altered quantities by establishing algebraic proportions.
Tahmini Süre:1m 15s
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Konular

Ratios, Rates, and Proportions

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